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On the Training of Infinitely Deep and Wide ResNets

On the Training of Infinitely Deep and Wide ResNets

Talk at Optimization and Statistical Learning 2023, updated in 2024.

The associated paper is:

Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport
Raphaël Barboni (ENS-PSL), Gabriel Peyré (CNRS and ENS-PSL), François-Xavier Vialard (LIGM)
https://arxiv.org/abs/2403.12887

Gabriel Peyré

January 15, 2023
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  1. On the Training of Infinitely Deep and Wide ResNets Gabriel

    Peyré É C O L E N O R M A L E S U P É R I E U R E François-Xavier Vialard Raphaël Barboni
  2. ResNet-34 <latexit sha1_base64="hGNaHRogJoszxpvRv/VDNWjykms=">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</latexit> image ResNet-type Architectures [He et al’ 16]

  3. ResNet-34 <latexit 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image <latexit 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dimension

    ResNet-type Architectures [He et al’ 16] x ↦ x + v(x) skip connexions
  4. ResNet-34 <latexit 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dimension

    ResNet-type Architectures [He et al’ 16] <latexit 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x0 <latexit 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x1 <latexit 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x0 <latexit 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x1 <latexit 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x0 <latexit 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x1 xS x2S x4S Makes « infinite depth » non-degenerate (ODE) → Enable initialization (identity map) → v = 0 Makes « infinite width » theory tractable. → Intuition: infinite width and depth flow avoids local minima. x ↦ x + v(x) skip connexions
  5. Infinite Width 2-layer Perceptrons vθ (x) := 1 q q

    ∑ k=1 σ(⟨θin k , x⟩)θout k f(θ) := 1 N ∑ i ℓ(vθ (xi ), yi ) Non-convex <latexit 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✓in <latexit 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✓out Simple yet universal
  6. Infinite Width 2-layer Perceptrons vθ (x) := 1 q q

    ∑ k=1 σ(⟨θin k , x⟩)θout k f(θ) := 1 N ∑ i ℓ(vθ (xi ), yi ) Non-convex vμ (x) := ∫ σ(⟨θin, x⟩)θoutdμ(θ) ℱ(μ) := ∑ i ℓ(vμ (xi ), yi ) Convex <latexit 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✓in <latexit 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✓out μ = 1 q ∑ k δθk q → + ∞ Simple yet universal measure parametri zation
  7. Infinite Width 2-layer Perceptrons vθ (x) := 1 q q

    ∑ k=1 σ(⟨θin k , x⟩)θout k f(θ) := 1 N ∑ i ℓ(vθ (xi ), yi ) Non-convex vμ (x) := ∫ σ(⟨θin, x⟩)θoutdμ(θ) ℱ(μ) := ∑ i ℓ(vμ (xi ), yi ) Convex Theorem: [Chizat-Bach 2018] if is large enough, the dynamic converges to global minima. q <latexit 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✓in <latexit sha1_base64="opFtcb5uSe0ZdHDGuvxJaagaDDw=">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</latexit> ✓out μ = 1 q ∑ k δθk q → + ∞ Simple yet universal measure parametri zation
  8. <latexit 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x0 <latexit 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x1 <latexit 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x(0) <latexit

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  9. <latexit 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x0 <latexit 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x(0) <latexit

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  10. Training with Infinite Depth and Width <latexit 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<latexit 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    x <latexit 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✓in <latexit sha1_base64="opFtcb5uSe0ZdHDGuvxJaagaDDw=">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</latexit> ✓out vμ (x) Infinite width: measure parameterization. … <latexit 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A + <latexit 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xi <latexit sha1_base64="s53dkvC9jV3TNtFIhxKVzVzByY8=">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</latexit> B + vμ(x) := ∫ σ(⟨θin, x⟩)θoutdμ(θ) vμ1 vμ2 Φμ μ := (μ1 , …, μS )
  11. Training with Infinite Depth and Width <latexit 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<latexit 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    x <latexit 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✓in <latexit sha1_base64="opFtcb5uSe0ZdHDGuvxJaagaDDw=">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</latexit> ✓out vμ (x) Infinite width: measure parameterization. … <latexit 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A + <latexit 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xi <latexit 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B + vμ(x) := ∫ σ(⟨θin, x⟩)θoutdμ(θ) vμ1 vμ2 Φμ μ := (μ1 , …, μS ) Infinite depth: ODE in depth. dx ds (s) = vμs (x(s)) <latexit 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A <latexit 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xi <latexit 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B Φμ μ := (μs )s∈[0,1]
  12. Training with Infinite Depth and Width <latexit 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<latexit 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    x <latexit 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✓in <latexit sha1_base64="opFtcb5uSe0ZdHDGuvxJaagaDDw=">AABE/3ictVxbbxTJFS42tw25scljFKk3XiI2IsQQcpFWkRY8xngxYJixYZcBNJf2eKA9Pcz0DIbZeYjyY6K8RFHylKf8jvyASMlT/kLOpaqreqa6T7VDaNmurq7vnFOnq06dc6qa7jgZTrPNzX+ce+8rX/3a17/x/jfPf+vb3/nu9y588P3DaTqb9OKDXpqkk8fdzjROhqP4IBtmSfx4PIk7J90kftR9uYXPH83jyXSYjlrZm3H89KQzGA2Phr1OBlXPL/xo0SYii0ncX7az42eLdhafZot0li2Xy+cXNjavbNK/aL1wVRc2lP63n37w4T9VW/VVqnpqpk5UrEYqg3KiOmoK1xN1VW2qMdQ9VQuom0BpSM9jtVTnATuDVjG06EDtS/g9gLsnunYE90hzSugecEngZwLISF0ETArtJlBGbhE9nxFlrC2jvSCaKNsb+NvVtE6gNlPHUCvhTMtQHPYlU0fqN9SHIfRpTDXYu56mMiOtoOSR06sMKIyhDst9eD6Bco+QRs8RYabUd9Rth57/i1piLd73dNuZ+jdJeRGuSDV179OcQkfNiX5Eb3MGz1ieBDgPgEKs+4il16TrE+r9CNovoP4eXEsqGZ104VpQ7bISuQWXD7klInfg8iF3ROQeXD7knojch8uH3NdIxE5I5358Ey4fvilyfgCXD/lARD6Ey4d8KCIP4fIhD0XkF3D5kF+IyFtw+ZC3ROQduHzIOyKyBZcP2RKRB3D5kAcichsuH3JbI8tn6gSulOgMhVl5A8pFHmgpEqi5Icp3k6yjD3szYE73SrDyrG7AXz+2EaDTuAS7HTDujkqw8sjbARvpx8q26DatJj7sbRG7CyPAj90VsZ+pFyXYzwJm2ssSrDzX9qCdHytb37tw58feFbH3oOTHymvUfajxY+8HrBjjEuy+iH2gXpVgQ6z+pAQr2/0m2BU/Vl6nWtDejw2xprMSrGxPD8GD8WPl1eoR1Pqxj0TsY3Vagn0sYj8H6+7Hfh6wwr4twZo19jytIAPyR2KYsVXUOvmsxNIYqHUE/km+tiTkG3ehXsIMcsyAMCciYidH7AQi9nLEXrBc09yOTsnflbk0c0QzENHN1yYsZWL7ft4eS0kAopEjGiuIKo8U37Xpy5y8C1MjIbN85cJSSJ/S3H5jKdbjodryGsT9AoLH9jGN/MsULWEEhZqqonacr/GMjOi+CvGaojfTS8NDxmW5VXBRpyKq60F1RdQbD+qNiJp5UDMRNfeg5iLKznwX1w4YAVb/+C4WdMcjgH3k8isCr+AGrDq3YY5GMH72wQt8SDX34W+TYm/pqpIMo3lcJzHL8bRgiSdQWqgNqLdRYYPi64RmWAySccv7OsbHO8xtLPScYyu8zFfyKM+YhNMZkjyDnA56ixHNp3p07lDNkrw7LtXD387nvSnVw2+TxpfkxXOpHj7T0mdnkL2lsa0zYJswm8Za+7ZclwbnX5iGKZ+nVRctLr7VEz1mkN5pTfq7+s3snuG9bFGJ9WPL9WhMnf5NC/2rQ8PqeerouR4V9J7Y6zWlqHZPRjruteW6MqS0io60HPau7pvBNn39Zky5Ho198Li2KOZeOOW6o3ec98aW69E4VJz3XJInb8r1aAzonvVhy/VoYLalo+N8W65r2VEDHDvbcl2rPqIsMOaAeMxzjfWKJuQnzTS1IfkH1dka1+dfX8cwZ/MsjxGqKVnftpxON1/LqiUy/kIMVi2rKQf6FzPHByvSWKhrYnzFMmSF9X2djl3jUfN7oMUIZj/vAUg58wQkNDkJtN4JULwqRl3FnhncNRGHo+RoBdXWtZnoLVq+nDUq1j2nWikus721emyTvZ7S2BuTT7hHmpX0sFf6hssoShraK2hIpldHd2/1fC1qf1PEjVcQ43yk9WhHiHfSquNUn9abjo4v6l2eDC7e87HjF7PNR9raYMyTki1CWap4uu1MHsmtw3X1srI5bn4W0RtFezUnqzGkHampGIWabDF74wu6t7QPaE8OeTCNHrzHSFMZK941wyw65tMjsqiuvZV4o75Mho7LU7K6xh5XowcOeuBB149xtmDFuAelFsQMB3DXCohyzue6SknjE/WzfHc0pTdYHdEnBQtpaLC9iQsWsirKPi5QeQ1oHA0cpYfTWKVj8O01SnLU75PHxq5Fy3+Rdm7N/naHxnj5aC7PxPSJ6zXiGtGs4V1dvlvlwBIsvE+ukf9a3UvkV4cj2lCJ6zOHM+tlRDv+MUWwY/KME5pt0uwotnbzU6tPDKd9ZfbOcTc7JQsZkf2LYH1KaUxG9OOeHTA76GwRErKRIXZnmHs3Pl9nKI4x68cNFZ9qsOMtJls2I/6Grju7pjQWOWLgdWC5MraNTvbIF4yJ60Rbdzu3q1cfRNpzEu4oYYp2rFwi/h/Tb/NjxsnG2ohADeMbmGpb53sfKcUsqKMOrfLVNsi0daX8KJfhmZbarn9Wpo8KkjUo4kJ5cLXuA+ce3TMvHCUTknu61obX0apsLlIer+gRe3tEUTzb/YFegVHuy7RKbtCca9MoGcAoyPIowrSVssirfKt5FamH0Z7+X6hbXRe1hhQjZTO4rCEpvx9TtOZKmcCo5vH7kmaTX+uTlVbVfEY0Fk+cufwl1H4Iv43c5j6MTrdgFW7SGGAK9s5qhGuitRZhvG4WeJmRaWjZe8vPjknTyq05S3zN1s3G2PPaVPZp1JzqrIUpn4XGC4fGi0Adtmiv0WrR1BtL9FyMLVp6tzKUXx1urRqUZyJl2SMzqGGAlG4sFUa1L1KVY3yDeivS2hRpdWC2ursB7pwPQfrn+urs/jJf3SN1i3ybHnlgHL/0aZYOyecytdWRGlNAzte1fXVnf5tqkHuXLChS5nOcOGN416lH1zKX9Cd6ZUvJzluLYM4tvdZtjI1tU/kXa8gTmhNTmpcGcZ1axFp+V45oxSJdcXyOiDL/HfKp2O+ojpnd1vadRAV/wsabPKssL44URqR/KfO2uxa97jrxa0Qx4Ux7112gVf8NIwXGmEyC37Oc0hvCVY53Etij7ZL9XLdTvIs3ciS6QlIv1G8DbAxHvXasu2PL9Nj07afQErVu37qvhcwvCeYo8TvLjl6HVrUT7aMuVu7PRqujV7nifZUeZit8rT5m1MaNLGyUV8S01SfBXFiielwYE8KlXi/qyF9P8joy8+5UKGXT2lAuZhrYxhxTvCSdA0WEz7u75PXmPhb60V2j1yWsS41rJEqYjUt1fsC1tJiVilYiJLdeWpMSZz0qWy8sD3fVsHacLWVMVjBRUu6GW7t9aBeiFTkbwxR6ik/2lsWJLs1P4MLfkfJFiYZjSA6xCX7uDbWltt/BqYhXusyZzYhq0Cb0V2Lwju5nsUW1jl451F36IRzCeQxB15L0Q1pR68rOlGXJXerh9F+TNZioWJTetqzfB5eL3JN1TnX6MyQLJ/dmqMw3OXX7YjiE9KTIJZwP729IvThS5tumen0w1OUeFDnU4WHOM4S9c9u6Pi+XU7W+1rmE8uB1wOy8GBzuAJbHLLZdiIWaOG/k3XNA63BUQd2sFv9rPwwfy6k+r1BuU/rm7EXAW+d2sc7Mol9cf85YbiGjuZxjOM807531mvz82P+Lar2p1OnNu6ePfqkdA4bXQnE+VJaO8e4osvKGUsH9AZ8MqfqP+vs5+auEVzmNMjnqUDL7FeXUTAuZmvny0tc78yxEJkunTKYiNRtPNOlk7JbaVbfgZyv3AOueEuVvKvkvYv3f0fah9oish8mmcwahTXUxZUHsblqf7u052jKJ8Uwvn/FtQQ3uie9RLZ73vUft8cxvq9C38i9JeK7fVanqFyKT1V0+O6+60IPiDhzngsz3vhGdqedsFp9AOwnYY+RzVBwpma+fF4ToU1y4KumCEGa0VFHueil36UxSXEK7W+hbj0b4WO/0474Dns/v5NmlSP2c6jp6dcCVWpJq3yPVE8oMdEn/mxCh/VJdhr+Xddkv6f6apFN6B0WJTp1n1SfBlt5xYb9mvEh5MJOpm+t2KUX1dvewOhPbKOXCJ96r8YMK/MCRsklv6yXF3RNVnTucVdCcaZnc/dyRMnlP1gNGs518fFTHz/MKXvOA/t8pRd9xJN0BWbqUbY9oP29C9BKtm22Sns9VVudtb1dIa77aZJr2ZKUdB+aMZPWeQKLHXfns53OQUq4mLqHjznU+kSmdFhl6KcnzcxxwGqIT0Fu5ryE9lajMRElmAV8izwNkmQfQORKkORIpDERJtH14fmHj6ur/9bFeOLx25eqvrlx/cH3j05v6/wF5X/1Q/VhdgrXv1+pTGP/76gA4/V79Uf1F/bXxu8YfGn9q/JmbvndOY36gCv8af/svMcZNUw==</latexit> ✓out vμ (x) Infinite width: measure parameterization. … <latexit 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A + <latexit 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xi <latexit 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B + vμ(x) := ∫ σ(⟨θin, x⟩)θoutdμ(θ) vμ1 vμ2 Φμ Training: minμ ℱ(μ) := 1 N ∑N i=1 ∥BΦμ (Axi) − yi∥2 μ := (μ1 , …, μS ) Infinite depth: ODE in depth. dx ds (s) = vμs (x(s)) <latexit 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A <latexit 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xi <latexit 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B Φμ μ := (μs )s∈[0,1]
  13. W2 (μ, ν)2 := min T q ∑ k=1 ∥θk

    − ξT(k) ∥2 Optimal Transport (Wasserstein) Distance ∥θi − ξj ∥2 θi ξj T Monge 1784
  14. W2 (μ, ν)2 := min T q ∑ k=1 ∥θk

    − ξT(k) ∥2 Optimal Transport (Wasserstein) Distance ∥θi − ξj ∥2 θi ξj T = inf T♯ μ=ν ∫ ∥θ − T(θ)∥2dμ(θ) μ ν T Monge 1784
  15. W2 (μ, ν)2 := min T q ∑ k=1 ∥θk

    − ξT(k) ∥2 Optimal Transport (Wasserstein) Distance ∥θi − ξj ∥2 θi ξj T = inf T♯ μ=ν ∫ ∥θ − T(θ)∥2dμ(θ) μ ν T Monge 1784 General measures: Kantorovitch relaxation Approximation by discrete measures or Kantorovitch 1942
  16. Wasserstein Gradient Flows Minimization over measures: min μ ℱ(μ) μ(t+τ)

    := arg min μ 1 2τ W2 2 (μ(t), μ) + ℱ(μ) ∂μ(t) ∂t = div(μ(t) ∇W ℱ(μ(t)) ) τ → 0 Wasserstein gradient ∇W ℱ(μ) = ∇ℝd f(θ) μ(t) μ(t+τ) μ(t+2τ) … μ = 1 q ∑ k δθk f(θ(t)) := ℱ(μ(t))
  17. Wasserstein Gradient Flows Minimization over measures: min μ ℱ(μ) μ(t+τ)

    := arg min μ 1 2τ W2 2 (μ(t), μ) + ℱ(μ) ∂μ(t) ∂t = div(μ(t) ∇W ℱ(μ(t)) ) τ → 0 Wasserstein gradient … ∇W ℱ(μ) = ∇ℝd[ δℱ δμ (μ)] ∇W ℱ(μ) = ∇ℝd f(θ) μ(t) μ(t+τ) μ(t+2τ) … μ = 1 q ∑ k δθk f(θ(t)) := ℱ(μ(t))
  18. Wasserstein Gradient Flows Minimization over measures: min μ ℱ(μ) μ(t+τ)

    := arg min μ 1 2τ W2 2 (μ(t), μ) + ℱ(μ) ∂μ(t) ∂t = div(μ(t) ∇W ℱ(μ(t)) ) τ → 0 Wasserstein gradient … ∇W ℱ(μ) = ∇ℝd[ δℱ δμ (μ)] Felix Otto David Kinderlehrer Richard Jordan ℱ(μ) = ∫ log( dμ dx )dμ Entropy: ∂μ(t) ∂t = Δμ(t) ∇W ℱ(μ) = ∇ℝd f(θ) μ(t) μ(t+τ) μ(t+2τ) … μ = 1 q ∑ k δθk f(θ(t)) := ℱ(μ(t))
  19. Conditional Wasserstein Flows s 0 1 Depth Conditional Wasserstein distance:

    𝒲 2 2 (μ, ν) := ∫1 0 W2 2 (μs , νs )ds W2 W2 W2
  20. Conditional Wasserstein Flows min μ ℱ(μ) μ(t+τ) := arg min

    μ 1 2τ 𝒲 2 2 (μ(t), μ) + ℱ(μ) ResNet training: s 0 1 Depth Optimization t μ(t) μ(t+τ) μ(t+2τ) Conditional Wasserstein distance: 𝒲 2 2 (μ, ν) := ∫1 0 W2 2 (μs , νs )ds W2 W2 W2
  21. Conditional Wasserstein Flows ∂μ(t) s ∂t = div(μ(t) s ∇

    𝒲 ℱ(μ(t))s ) τ → 0 min μ ℱ(μ) μ(t+τ) := arg min μ 1 2τ 𝒲 2 2 (μ(t), μ) + ℱ(μ) ResNet training: s 0 1 Depth Optimization t μ(t) μ(t+τ) μ(t+2τ) Conditional Wasserstein distance: 𝒲 2 2 (μ, ν) := ∫1 0 W2 2 (μs , νs )ds W2 W2 W2
  22. Conditional Wasserstein Flows ∂μ(t) s ∂t = div(μ(t) s ∇

    𝒲 ℱ(μ(t))s ) τ → 0 min μ ℱ(μ) μ(t+τ) := arg min μ 1 2τ 𝒲 2 2 (μ(t), μ) + ℱ(μ) ResNet training: s 0 1 Depth Optimization t μ(t) μ(t+τ) μ(t+2τ) Finite width μ(t) s = 1 q ∑ k δθ(t) s,k f(θ(t)) := ℱ(μ(t)) dθ(t) dt = − ∇f(θ(t)) Conditional Wasserstein distance: 𝒲 2 2 (μ, ν) := ∫1 0 W2 2 (μs , νs )ds W2 W2 W2
  23. Conditional Wasserstein Flows ∂μ(t) s ∂t = div(μ(t) s ∇

    𝒲 ℱ(μ(t))s ) τ → 0 min μ ℱ(μ) μ(t+τ) := arg min μ 1 2τ 𝒲 2 2 (μ(t), μ) + ℱ(μ) ResNet training: s 0 1 Depth Optimization t μ(t) μ(t+τ) μ(t+2τ) Finite width μ(t) s = 1 q ∑ k δθ(t) s,k f(θ(t)) := ℱ(μ(t)) dθ(t) dt = − ∇f(θ(t)) Conditional Wasserstein distance: 𝒲 2 2 (μ, ν) := ∫1 0 W2 2 (μs , νs )ds W2 W2 W2 Theorem: under smoothness and growth conditions on , exists and is unique σ (s, t) ↦ μ(t) s
  24. Polyak-Łojasiewicz Condition <latexit sha1_base64="c84IgZCfgfqve0vx8DSH4w339nY=">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</latexit> Polyak- Lojasiewicz inequality: 0 ≤ mf(θ)

    ≤ ∥∇f(θ)∥2 2 no spurious stationary points. → Example: with strongly convex f(θ) = g(Aθ) g
  25. Polyak-Łojasiewicz Condition <latexit sha1_base64="c84IgZCfgfqve0vx8DSH4w339nY=">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</latexit> Polyak- Lojasiewicz inequality: 0 ≤ mf(θ)

    ≤ ∥∇f(θ)∥2 2 no spurious stationary points. → Example: with strongly convex f(θ) = g(Aθ) g
  26. Polyak-Łojasiewicz Condition <latexit sha1_base64="c84IgZCfgfqve0vx8DSH4w339nY=">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</latexit> Polyak- Lojasiewicz inequality: 0 ≤ mf(θ)

    ≤ ∥∇f(θ)∥2 2 no spurious stationary points. → Example: with strongly convex f(θ) = g(Aθ) g <latexit sha1_base64="kI5fXwi7ymCUn+8sWlWuAyRczCA=">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</latexit> [Polyak 1963] <latexit 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Theorem: f(θ(t)) ≤ e−mt f(θ(0)) d dt [f(θ(t))] = ⟨∇f(θ(t)), d dt θ(t)⟩ = −∥∇f(θ(t))∥2 ≤ − mf(θ(t)) Grönwall's inequality → For , · θ = − ∇f(θ)
  27. Polyak-Łojasiewicz Condition <latexit sha1_base64="c84IgZCfgfqve0vx8DSH4w339nY=">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</latexit> Polyak- Lojasiewicz inequality: (Wasserstein P-Ł) 0

    ≤ mℱ(μ) ≤ ∥∇ 𝒲 ℱ(μ)∥2 L2(μ) ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) Theorem: [L. D.Schiavo et al 2023] ⟹ 0 ≤ mf(θ) ≤ ∥∇f(θ)∥2 2 no spurious stationary points. → Example: with strongly convex f(θ) = g(Aθ) g <latexit sha1_base64="kI5fXwi7ymCUn+8sWlWuAyRczCA=">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</latexit> [Polyak 1963] <latexit 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Theorem: f(θ(t)) ≤ e−mt f(θ(0)) d dt [f(θ(t))] = ⟨∇f(θ(t)), d dt θ(t)⟩ = −∥∇f(θ(t))∥2 ≤ − mf(θ(t)) Grönwall's inequality → For , · θ = − ∇f(θ)
  28. Obstruction for Global P-Ł for Neural ODEs <latexit sha1_base64="IxeOLUkTvceeSFRCo84cOj0XSVA=">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</latexit> Linear

    ResNet, time-independant weights: <latexit sha1_base64="t7eG4Pjr6u0TNKVBuMuFuZ5awqE=">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</latexit> ˙ x = ✓x <latexit 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✓(x) = e✓x <latexit sha1_base64="waTHm3fQpdQQpFlwopY3HfEpmdU=">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</latexit> For yj = xi, i.e. learning Id: <latexit sha1_base64="XY6WqX8kPJU7sq5nmngGZEW6eVI=">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</latexit> f(✓) , ||e✓ + Id||2 d dt θ(t) = − ∇f(θ(t))
  29. Obstruction for Global P-Ł for Neural ODEs <latexit sha1_base64="IxeOLUkTvceeSFRCo84cOj0XSVA=">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</latexit> Linear

    ResNet, time-independant weights: <latexit sha1_base64="t7eG4Pjr6u0TNKVBuMuFuZ5awqE=">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</latexit> ˙ x = ✓x <latexit 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✓(x) = e✓x <latexit 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⇡ <latexit sha1_base64="waTHm3fQpdQQpFlwopY3HfEpmdU=">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</latexit> For yj = xi, i.e. learning Id: <latexit sha1_base64="XY6WqX8kPJU7sq5nmngGZEW6eVI=">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</latexit> f(✓) , ||e✓ + Id||2 d dt θ(t) = − ∇f(θ(t)) Proposition: If θ(0) = Udiag(z(0) 1 , z(0) 2 , …)U* then θ(t) = Udiag(z(t) 1 , z(t) 2 , …)U* where is a gradient flow of z(t) ∈ ℂ ˜ f(z) := |ez + 1|
  30. Obstruction for Global P-Ł for Neural ODEs <latexit sha1_base64="IxeOLUkTvceeSFRCo84cOj0XSVA=">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</latexit> Linear

    ResNet, time-independant weights: <latexit sha1_base64="t7eG4Pjr6u0TNKVBuMuFuZ5awqE=">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</latexit> ˙ x = ✓x <latexit 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✓(x) = e✓x <latexit 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⇡ <latexit sha1_base64="waTHm3fQpdQQpFlwopY3HfEpmdU=">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</latexit> For yj = xi, i.e. learning Id: <latexit sha1_base64="XY6WqX8kPJU7sq5nmngGZEW6eVI=">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</latexit> f(✓) , ||e✓ + Id||2 d dt θ(t) = − ∇f(θ(t)) Proposition: If θ(0) = Udiag(z(0) 1 , z(0) 2 , …)U* then θ(t) = Udiag(z(t) 1 , z(t) 2 , …)U* where is a gradient flow of z(t) ∈ ℂ ˜ f(z) := |ez + 1| Problem: does not satisfy a global P-Ł ˜ f For Im(z(0)) = 0[2π], Re(z(t)) → − ∞ if (not invertible) ⇔ θ(0) = Id, eθ(t) → 0
  31. Local P-Ł Condition Theorem: [L. D.Schiavo et al 2023] If

    for , 𝒲 (μ(0), μ) ≤ R <latexit sha1_base64="/aDLqZ3M48mYQOV8H1W/SqnogP0=">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</latexit> f(✓) 0 ≤ mℱ(μ) ≤ ∥∇ 𝒲 ℱ(μ)∥2 L2(μ) and , then ℱ(μ(0)) ≤ mR2 4 ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) 𝒲 2 (μ(0), μ(t)) ≤ R μ(t) μ(0)
  32. Local P-Ł Condition Theorem: [L. D.Schiavo et al 2023] If

    for , 𝒲 (μ(0), μ) ≤ R <latexit sha1_base64="/aDLqZ3M48mYQOV8H1W/SqnogP0=">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</latexit> f(✓) 0 ≤ mℱ(μ) ≤ ∥∇ 𝒲 ℱ(μ)∥2 L2(μ) and , then ℱ(μ(0)) ≤ mR2 4 ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) 𝒲 2 (μ(0), μ(t)) ≤ R μ(t) μ(0)
  33. Local P-Ł Condition Theorem: [L. D.Schiavo et al 2023] If

    for , 𝒲 (μ(0), μ) ≤ R <latexit sha1_base64="/aDLqZ3M48mYQOV8H1W/SqnogP0=">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</latexit> f(✓) 0 ≤ mℱ(μ) ≤ ∥∇ 𝒲 ℱ(μ)∥2 L2(μ) and , then ℱ(μ(0)) ≤ mR2 4 ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) 𝒲 2 (μ(0), μ(t)) ≤ R μ(t) μ(0) Questions: conditions on initialization μ(0) data (xi , yi )N i=1 to ensure a local P-Ł Condition?
  34. Local P-Ł Condition for ResNet Feature kernel: K[μ]i,j := ∫

    σ(⟨θin, xi ⟩)σ(⟨θin, xj ⟩)dμ(θ) Identity initializations (« FixUp »): μ(0) s (θ) = δ0 (θout) ⊗ ˜ μ(0)(θin)
  35. Local P-Ł Condition for ResNet Feature kernel: K[μ]i,j := ∫

    σ(⟨θin, xi ⟩)σ(⟨θin, xj ⟩)dμ(θ) Identity initializations (« FixUp »): μ(0) s (θ) = δ0 (θout) ⊗ ˜ μ(0)(θin) R = Cκ Theorem: If is bounded and σ′  κ := λmin (K[ ˜ μ(0)]) > 0 then Local P-Ł holds for Corrolary: If , then ℱ(μ(0)) ≤ (Cκ)3 4N ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) M2 (μ) := ∫ ∥θ∥2dμ(θ) m = CκN−1 C ∝ e−∥σ′  ∥∞ M2 ( ˜ μ(0))
  36. Local P-Ł Condition for ResNet Feature kernel: K[μ]i,j := ∫

    σ(⟨θin, xi ⟩)σ(⟨θin, xj ⟩)dμ(θ) Identity initializations (« FixUp »): μ(0) s (θ) = δ0 (θout) ⊗ ˜ μ(0)(θin) δ := min i≠j ∥xi − xj ∥ > 0 has a « nice » density. ˜ μ(0) <latexit 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xi <latexit 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> <latexit 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yi Φμ (positivity of ) K[μ] m := λmin (K[ ˜ μ(0)]) > 0 ⟹ R = Cκ Theorem: If is bounded and σ′  κ := λmin (K[ ˜ μ(0)]) > 0 then Local P-Ł holds for Corrolary: If , then ℱ(μ(0)) ≤ (Cκ)3 4N ℱ(μ(t)) ≤ e−mt ℱ(μ(0)) M2 (μ) := ∫ ∥θ∥2dμ(θ) m = CκN−1 C ∝ e−∥σ′  ∥∞ M2 ( ˜ μ(0))
  37. Enforcing Convergence via Lifting & Discretization Neural ODE constraint: is

    a diffeomorphism. Φμ universality (zero training loss) requires lifting. → Lifting: Φμ · x = vθs (x) A B A = (Idd ,0) B = (0,Idd ) ℝ2d ℝ2d ℝd ℝd At initialization → ℱ(μ(0)) = 1 N ∑ i ∥yi ∥2 ≤ (Cκ)3 4N
  38. Enforcing Convergence via Lifting & Discretization Neural ODE constraint: is

    a diffeomorphism. Φμ universality (zero training loss) requires lifting. → Lifting: Finite width: random sampling of neurons. q = O(N2p) Finite depth: extending [Marion, Wu, Sander, Biau, ?ICLR? 2023] Φμ · x = vθs (x) A B A = (Idd ,0) B = (0,Idd ) ℝ2d ℝ2d ℝd ℝd At initialization → ℱ(μ(0)) = 1 N ∑ i ∥yi ∥2 ≤ (Cκ)3 4N
  39. Open Problems! TODO Global convergence: for generic initialization. Unsupervised learning:

    normalizing flows. Hard even for linear networks … → Use e.g. Wasserstein loss. → Beyond minimum separation : δ > 0 smooth density of data/labels. → <latexit sha1_base64="Jm6pcXy4mbMQAGLxbG+Df1aMWrA=">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</latexit> > Φμ
  40. Open Problems! TODO Global convergence: for generic initialization. Unsupervised learning:

    normalizing flows. Hard even for linear networks … → Use e.g. Wasserstein loss. → Beyond minimum separation : δ > 0 smooth density of data/labels. → <latexit sha1_base64="Jm6pcXy4mbMQAGLxbG+Df1aMWrA=">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</latexit> > Φμ
  41. Open Problems! TODO Global convergence: for generic initialization. Unsupervised learning:

    normalizing flows. Hard even for linear networks … → Use e.g. Wasserstein loss. → <latexit sha1_base64="YhX57l/gdlHksS6L+bN+TenB9Bk=">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</latexit> ODEs <latexit sha1_base64="niytPyfrKEbB5sXTKqk1fyJ2pPQ=">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</latexit> Wasserstein flows <latexit 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(single point) <latexit 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(group of points) Transformers: treat activation as measures xi (0) xi (1) ∑ i δxi (0) ∑ i δxi (1) Beyond minimum separation : δ > 0 smooth density of data/labels. → <latexit 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> Φμ